Q: Does a handwritten will and request not to be kept alive artificially hold up in New Mexico?
A: How a will is written (by hand, typewriter, computer, etc.) is irrelevant. It depends on what the will says and how it is executed (signed) to determine if it is valid or not. However, a will has nothing to do with being kept alive artificially in New Mexico or anywhere else in the world.
A will disposes of stuff after you die.
What you are thinking of is a power of attorney (incorrectly called "a living will"). And, again, it depends on how it is executed (signed) to determine if it is valid or not; it is irrelevant how the words get on the paper.
With that knowledge, now tell us what you really want to know.
Facts?
Details?